Central suboptimal H ∞ filter design for linear time-varying systems with state and measurement delays

• Published in: International journal of systems science 2010-04, Vol.41 (4), p.411-421
• Main Authors: Basin, Michael, Shi, Peng, Calderon-Alvarez, Dario
• Format: Article
• Language: English
• Subjects: Adaptative systems; Applied sciences; Computer science; control theory; systems; Control system analysis; Control systems; Control theory. Systems; Delay; Exact sciences and technology; Filtering; Filtration; H-infinity control; Linear systems; Mathematical analysis; Modelling and identification; Optimization; robust filtering; time-delay systems
• ISSN: 0020-7721; 1464-5319
• DOI: 10.1080/00207720903045825

This article presents the central finite-dimensional H ∞ filters for linear systems with state and measurement delay that are suboptimal for a given threshold γ with respect to a modified Bolza-Meyer quadratic criterion including the attenuation control term with the opposite sign. In contrast to the results previously obtained for linear time delay systems, this article reduces the original H ∞ filtering problem to H 2 (optimal mean-square) filtering problem using the technique proposed in Doyle, Glover, Khargonekar, and Francis (1989 'State-space Solutions to Standard H 2 and H ∞ Control Problems', IEEE Transactions on Automatic Control, 34, 831-847). Application of the reduction technique becomes possible, since the optimal closed-form filtering equations solving the H 2 (mean-square) filtering problem have been obtained for linear systems with state and measurement delays. This article first presents the central suboptimal H ∞ filter for linear systems with state and measurement delays, based on the optimal H 2 filter from Basin, Alcorta-Garcia, and Rodriguez-Gonzalez (2005, 'Optimal Filtering for Linear Systems with State and Observation Delays', International Journal of Robust and Nonlinear Control, 15, 859-871), which consists, in the general case, of an infinite set of differential equations. Then, the finite-dimensional central suboptimal H ∞ filter is designed in case of linear systems with commensurable state and measurement delays, which contains a finite number of equations for any fixed filtering horizon; however, this number still grows unboundedly as time goes to infinity. To overcome that difficulty, the alternative central suboptimal H ∞ filter is designed for linear systems with state and measurement delays, which is based on the alternative optimal H 2 filter from Basin, Perez, and Martinez-Zuniga (2006, 'Alternative Optimal Filter for Linear State Delay Systmes', International Journal of Adaptive Control and Signal Processing, 20, 509-517). In all cases, the standard H ∞ filtering conditions of stabilisability, detectability and noise orthonormality are assumed. Finally, to relax the standard conditions, this article presents the generalised versions of the designed H ∞ filters in the absence of the noise orthonormality. The proposed H ∞ filtering algorithms provide direct methods to calculate the minimum achievable values of the threshold γ, based on the existence properties for a bounded solution of the gain matrix equation. Numerica